Fourier transform of distributions




The generalization of the concept of Fourier transform from suitable function to distributions.



(Fourier transform of tempered distributions)

For nn \in \mathbb{N}, let u𝒮( n)u \in \mathcal{S}'(\mathbb{R}^n) be a tempered distribution. Then its Fourier transform u^𝒮( n)\hat u \in \mathcal{S}'(\mathbb{R}^n) is defined by

𝒮( n) u^ f u,f^, \array{ \mathcal{S}(\mathbb{R}^n) &\overset{\hat u}{\longrightarrow}& \mathbb{C} \\ f &\mapsto& \langle u, \hat f\rangle } \,,

where on the right f^\hat f is the ordinary Fourier transform of the function ff (an element of the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n)).

(e.g. Hörmander 90, def. 7.1.9)



The operation of Fourier transform of tempered distributions (def. ) induces a linear isomorphism of the space of temptered distributions with itself:

()^:𝒮( n)𝒮( n) \widehat{(-)} \;\colon\; \mathcal{S}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathcal{S}'(\mathbb{R}^n)

(e.g. Melrose 03, corollary 1.1)


(Fourier transform of compactly supported distributions)

If u𝒟( n)( n)u \in \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n) happens to be a compactly supported distribution, regarded as a tempered distribution, then its Fourier transform according to def. is a smooth function

u^C ( n) \hat u \in C^\infty(\mathbb{R}^n)

given by

u^(k)u(exp(i(),2πk)), \hat u(k) \;\coloneqq\; u\left( \exp(-i \langle (-), 2 \pi k \rangle) \right) \,,

where ,\langle -,-\rangle denotes the canonical inner product on n\mathbb{R}^n.

This is well-defined also on complex numbers, which makes it an entire holomorphic function (by the Paley-Wiener-Schwartz theorem), called the Fourier-Laplace transform.

(e.g. Hörmander 90, theorem 7.1.14)

(This plays a role for instance in the Paley-Wiener-Schwartz theorem.)


The Fourier transform (def. ) of the convolution of distributions of a compactly supported distribution u 1u_1 \in \mathcal{E}' with a tempered distribution u 2𝒮u_2 \in \mathcal{S}' is the product of distributions of their separate Fourier transforms:

u 1u 2^=u^ 1u^ 2. \widehat {u_1 \star u_2} \;=\; \hat u_1 \hat u_2 \,.

(Here, by prop. , u^ 1\hat u_1 is just a smooth function, so that the product on the right is just that of a distribution with a function.)

(Hörmander 90, theorem 7.1.15)


Last revised on April 2, 2020 at 09:32:47. See the history of this page for a list of all contributions to it.